Nephroid

A nephroid is a specific type of roulette, a curve generated by a point on a circle rolling along the inside of a larger circle. Specifically, it's the roulette formed when the radius of the rolling circle is exactly one-half the radius of the fixed circle. The resulting curve is a hypocycloid with two cusps.

The name "nephroid" originates from the Greek word "nephros" meaning "kidney," due to the curve's resemblance to a kidney shape. This shape is characterized by its two cusps pointing inwards and a smooth, curved outer boundary.

Mathematically, the nephroid can be described by parametric equations. Let R be the radius of the fixed circle and r the radius of the rolling circle (where r = R/2). The parametric equations for the nephroid are typically expressed in terms of a parameter t, representing the angle of rotation of the rolling circle. These equations define the x and y coordinates of points on the nephroid as functions of t.

The nephroid possesses interesting reflective properties. If parallel rays of light are directed into the concave side of the nephroid, they are reflected to a single point called the focus of the nephroid. This property has led to its study in optics.

The nephroid is related to other curves like the cardioid and the deltoid, all of which are hypocycloids with different ratios between the radii of the fixed and rolling circles. Understanding the properties of the nephroid contributes to a broader understanding of geometric curves and their applications.